Three isotopes of an element have mass numbers, m, (m + 1) and (m + 2). If the mean mass number is (m + 0.5) then which of the following ratios can be accepted for m, (m +1), (m + 2) in that order ?
  • a)
    1 : 1 : 1
  • b)
    4 : 1 : 1
  • c)
    3 : 2 : 1
  • d)
    2 : 1 : 1
Correct answer is option 'B'. Can you explain this answer?

Class 12 Question

Rakshit Dayma answered Aug 04, 2018
The mass number is the total number of protons and neutrons in the atom. But this is more of a maths question rather than chemistry. 

Let's assume there are x units of the isotope with mass number m

y units of the isotope with mass number m+1

z units of the isotope with mass number m+2


So the mean would be (xm + y(m+1) + z(m+2)) / (x+y+z)  [Mean = Total sum/Total quantity]
which is equal to m+0.5 [Given in question]


Solving this we get, 0.5(x - y - 3z) = 0=> x = y+3z     -----------equation (1)


Now we can put values of y and z and get x. There can be infinitely many values of x, y and z.

Here amongst the options, we can see that option (b) matches this criterion.


To be precise, if we take a constant k, then x=4k, y=k and c=k satisfy equation (1). 

We can also see that option (a) and option (c) don't satisfy equation (1)

So amongst the options, option (b) is the answer.

As a check, if we take other ratios which satisfy equation (1) they can also be an answer to the question.
e.g. 7:1:2 => x=7k, y=k, z=2k

Now the mean would be, (xm + y(m+1) + z(m+2))/(x+y+z)

Putting the values, 

(7km + k(m+1) + 2k(m+2))/(7k + k + 2k)

= (10 km + 5k)/(10k)

= m + 0.5

So 7:1:2 can also be a ratio which satisfies the given criterion.

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